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Question
Write the ratio in which the plane 4x + 5y − 3z = 8 divides the line segment joining the points (−2, 1, 5) and (3, 3, 2).
Solution
\[\text{ We know that the ratio in which the plane ax + by + cz + d = 0 divides the line segment joining} \left( x_1 , y_1 , z_1 \right)\text{ and } \left( x_2 , y_2 , z_2 \right)is\]
\[\frac{- \left( a x_1 + b y_1 + c z_1 + d \right)}{a x_2 + b y_2 + c z_2 + d}\]
\[\text{ Here } ,a = 4; b = 5; c = - 3; d = - 8; x_1 = - 2; y_1 = 1; z_1 = 5; x_2 = 3; y_2 = 3; z_2 = 2\]
\[\text{ So, the required ratio } \]
\[ = \frac{- \left( 4 \left( - 2 \right) + 5 \left( 1 \right) - 3 \left( 5 \right) - 8 \right)}{4 \left( 3 \right) + 5 \left( 3 \right) - 3 \left( 2 \right) - 8}\]
\[ = \frac{- \left( - 8 + 5 - 15 - 8 \right)}{12 + 15 - 6 - 8}\]
\[ = \frac{26}{13}\]
\[ = \frac{2}{1} \text{ or } 2 : 1\]
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